Concept mapping has a history of use in many disciplines as a formal or semi-formal diagramming technique. Concept maps have an abstract structure as typed hypergraphs, and computer support for concept mapping can associate visual attributes with node types to provide an attractive and consistent appearance. Computer support can also provide interactive interfaces allowing arbitrary actions to be associated with nodes such as hypermedia links to other maps and documents. This article describes a general concept mapping system that is open architecture for integration with other systems, scriptable to support arbitrary interactions and computations, and cutomizable to emulate many styles of map. The system supports collaborative development of concept maps across local area and wide area networks, and integrates with World-Wide Web in both client helper and server gateway roles. A number of applications are illustrated ranging through education, artificial intelligence, active documents...

Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can be used in conjunction with object-oriented modelling notations such as the Unified Modeling Language. This paper summarises the main syntax and semantics of spider diagrams. It also introduces inference rules for reasoning with spider diagrams and a rule for combining spider diagrams. This system is shown to be sound but not complete. Disjunctive diagrams are considered as one way of enriching the system to allow combination of diagrams so that no semantic information is lost. The relationship of this system of spider diagrams to other similar systems, which are known to be sound and complete, is explored briefly.

. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the "inference steps" of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive !-rule provides the mathematical basis for this step from schematic proofs to theoremhood. In ...

The effective use of visual languages requires a precise understanding of their meaning. Moreover, it is impossible to prove properties of visual languages like soundness of transformation rules or correctness results without having a formal language definition. Although this sounds obvious, it is surprising that only little work has been done about the semantics of visual languages, and even worse, there is no general framework available for the semantics specification of different visual languages. We present such a framework that is based on a rather general notion of abstract visual syntax. This framework allows a logical as well as a denotational approach to visual semantics, and it facilitates the formal reasoning about visual languages syntax and semantics for the visual languages VEX, Show and Tell, and Euler Circles. We demonstrate the semantics in action by proving a rule for visual reasoning with Euler Circles and by showing the correctness of a Show and Tell program.

The study of systems of communication may be divided into three parts: syntax, semantics and pragmatics. Accounts of the embedding of text-based languages in the computational processes of reasoners and communicators are relatively well developed; with accounts available for a spectrum of languages which ranges from the highly formalised and constrained, such as formal logics, to the highly informal and unconstrained natural languages used in everyday conversations. Analogies between diagrams and such textual representations of information are quite revealing about both similarities and differences and can provide a useful starting point for exploring the issues in a theory of diagrammatic communication. This paper sketches out a theory of diagrammatic communication, based upon recent studies of the syntactic, semantic and pragmatic component issues which such a theory must accommodate. In the context of this theory an exploration is made of the issues involved in answering...

Traditionally the dominant formalist school in mathematics has considered diagrams as merely heuristic tools. However, the last few years have seen a renewed interest in visualisation in mathematics and, in particular, in diagrammatic reasoning. This has resulted from the increasing capabilities of modern computers, the key role that design and modelling notations play in the development process of software systems, and the emergence of the first formal diagrammatic systems. Constraint diagrams are a diagrammatic notation for expressing constraints that can be used in conjunction with the Unified Modelling Language (UML) in object-oriented modelling. Recently, full formal semantics and sound and complete inference rules have been developed for Venn-Peirce diagrams and Euler circles. Spider diagrams emerged from work on constraint diagrams. They combine and extend Venn-Peirce diagrams and Euler circles to express constraints on sets and their relationships with other sets. The spider diagram system SD1 developed in this thesis extends the second Venn-Peirce system that Shin investigated, Venn II, to give lower bounds for the cardinality of the sets represented

Abstract. Spider diagrams are a visual notation for expressing logical statements. In this paper we identify a well known fragment of first order predicate logic, that we call ESD, equivalent in expressive power to the spider diagram language. The language ESD is monadic and includes equality but has no constants or function symbols. To show this equivalence, in one direction, for each diagram we construct a sentence in ESD that expresses the same information. For the more challenging converse we show there exists a finite set of models for a sentence S that can be used to classify all the models for S. Using these classifying models we show that there is a diagram expressing the same information as S. 1

Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which is on a fragment of the constraint diagram language, called spider diagrams, and constraint diagrams themselves. In the thesis, sound and complete systems of spider diagrams and constraint diagrams are presented and the expressiveness of the spider diagram language is established. 1

Abstract. Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They were designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Due to the lack of a linear ordering of symbols inherent in a diagrammatic language which expresses logical statements, some constraint diagrams have more than one intuitive meaning. We generalize, from an example based approach, to suggest a default reading for constraint diagrams. This reading is usually unique, but may require a small number of simple user choices.

We advance a theoretical framework...